42 



REV. HUGH MARTIN ON TRILINEAR CO-ORDINATES 



Theorem 9. 



A, is 

 A 2 is 

 A 3 is 



-1, 



-gh 



1-gW, 



9> 



h(g 2 - 



9 

 h 



1), 9(^-1) 



0. Therefore ; — 



A,A 2 A 3 is a straight line. 



B 1 B 2 B 3 » » 



^1 ^2^3 >> " 



Theorem 10. 



B 3 C 3 is 1+fgK h+fg, g + hf; meeting B C in 3, 

 C 3 A 3 is h+fg, l+fgh, f+gh; „ CA „ 33, 



A3B3 is g + hf, f+gh, l+fgh; „ AB „ 0, 



Therefore ;- 



BC, B,C„ B 2 C 2 , B3C3 meet in £,. 

 CA.CA.CA, C3A3 „ 33,. 

 AB, A,B„ A 2 B 2 , A3B3 „ <£,. 



Theorem 11. 



B C, B 4 C 4 meet in & 4 , viz., 

 CA, C 4 A 4 „ 33 4 , „ 



AB, A 4 B 4 



0, (f-gh) . (h+fg), -(f+gh).(g-hf) 



■(g + hf).(h-fg), 0, (g-hf).(f+gh) 



{h-fg) .{g+hf), -(h+fg).{f-gh), o 



Therefore Sl 4 33 4 © 4 is a straight line, viz., 



1 



(? + */)• (* -/SO' {h+fg).{f-ghV (f+gh).(g-hfy 



Theorem 12. 



BjC, A,B meet in A 5 , viz., j" — h, 



C 5> » 



C X A, B,C 

 A,B, C,A 



h, 1, cjA) 



!» /</» -g J 



BC,, CA, meet in A 6 , viz., ( —g, gh, 1 ] 

 CA„ AB, „ B c , „ 1 1, -A, hf 

 AB„ BC, „ C 6 , „ ( fg, 1, -/ J 



A 5 A 6 , B C meet in & 5 , viz,, 



B 5 B 6 ,CA 

 C s C fi ,AB 



*"5> " 



0, ^(1-A 2 ), -A(l-^) 



-/(i-n o, mi-/ 2 ) 



/(i-/). -?(i-/ 2 ), o 



0. Therefore ;- 



is a straight line, viz., 



1-/2 1_^2 1 _ A 5 



"7" T~' "A- 



Theorem 13. 



B C is 

 B 5 C 6 is 

 B 6 C 5 is 



1, 

 / f -l. 



%(W 2 ), 



0, 



g+hf 



o 



M+fg) 

 h+fg 



= 0. Therefore ;— 



B C, B 5 C 6 , B 6 C 5 meet in a point : 

 CA, C 5 A C , C 6 A 5 ,, ,, 



AB, A 5 B 6 , A 6 B 5 



©. 



